On the square peg problem
Abstract
We show that if $\gamma$ is a Jordan curve in $\mathbb{R}^2$ which is close to a $C^2$ Jordan curve $\beta$ in $\mathbb{R}^2$, then $\gamma$ contains an inscribed square. In particular, if $\kappa > 0$ is the maximum unsigned curvature of $\beta$ and there is a map $f$ from the image of $\gamma$ to the image of $\beta$ with $||f(x) - x|| < \frac{1}{10 \kappa}$ and $f \circ \gamma$ having winding number $1$, then $\gamma$ has an inscribed square of positive sidelength.
- Publication:
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arXiv e-prints
- Pub Date:
- March 2022
- DOI:
- 10.48550/arXiv.2203.02613
- arXiv:
- arXiv:2203.02613
- Bibcode:
- 2022arXiv220302613C
- Keywords:
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- Mathematics - Geometric Topology;
- Mathematics - Combinatorics;
- Mathematics - Metric Geometry;
- 53A04;
- 52C99
- E-Print:
- 11 pages, 3 figures